I'm writing a novel that is set in Vienna during the
period when it was occupied by, among others, the
Americans and the Soviets (1945-1955). The central
district (grey in the image below) belonged to all the
occupational powers.

In my novel there is a character (a native Austrian)
who seriously injures (on purpose) an American
soldier or officer.
After the crime he needs to hide somewhere. He
considers running to the Soviet zone in the hope
that it will make it more difficult for the
Americans to find him there. This leads to the
first question.

Question #1: Did the Soviets cooperate with
Americans when investigating serious crimes?

Let's say he runs to the Soviet zone. The Americans
somehow find this out and ask the Soviets to help
them find this criminal (or at least not interfere).
How likely are the Soviets to agree (provided that
the character committed a serious crime against the Americans and the
Soviets are not interested in him at all)?

Question #2: Whose police forces would perform a
large operation, if the need arose?

Now assume the Soviets help the Americans find the
protagonist actively or passively. It turns out the
protagonist has barricaded himself.

The only way to catch him is to storm that place.

Whose troops (Soviet, American) are more likely to
do this special operation given that

it happens in the Soviet-occupied part of the city,

the criminal is an Austrian,

whom the Americans want and the Russians don't

?

Note: Since I'm writing fiction, I do not need 100 %
historical accuracy. I only need believability.

It's still ficton, but have you seen the Third Man? That's got a similar plot to what you're describing, and takes place in occupied Vienna. You could perhaps go to Google Scholar and look for analyses of the film in historical journals. That could tell you which parts of the portrayal are accurate (or inaccurate)
– Ne MoFeb 7 '18 at 21:51

...if you can be bothered (I think fiction writers should, because fiction is the way the majority of people learn anything about the past at all, but that's by-the-by). If you just want believabilty, then just say whatever the Third Man said.
– Ne MoFeb 7 '18 at 21:58

If I wanted to research this topic, what would be a good starting point?
– Franz DrolligFeb 8 '18 at 7:45

These combined patrols were kept running for quite a while and continued even as the relations went sour.
But, just like in Germany, things deteriorated soon. The later years were marked by Soviet obstruction of almost anything, including policing.

As for concerns over Soviet obstructionism, a particularly noteworthy incident was the infamous, almost comical, Christmas Eve Gummiknüppel incident. Without warning on December 24, 1952, Soviet military detachments began fanning out through their zone confiscating the gendarmerie’s rubber truncheons. The Allies were at a loss to explain the behavior, but the U.S. ambassador cautioned that the Soviets “may be releasing trial balloon with intention push control further if unopposed this time. . . . In view customary Soviet technique of moving first against police in any area they intend to take over, we are concerned by possible seriousness Soviet action and propose strong reaction.” This response was not untypical. Any Soviet action against the police or gendarmerie was a matter of grave consternation. (p 187.)

Immediately after the war a criminal offender against Americans would have had a hard time hiding with the Soviets on these principles. Factors like communication difficulties between the allies would still make it a viable idea to try it in the east. At the end of the timeframe given, its very likely that the increasingly – perceived or real – erratic behaviour on the Soviet side made it quite likely that a request for cooperation would be denied or just ignored.